How does atmospheric temperature ranging between 10℃ to 55℃ with a regular interval of 5℃ affect the frequency of E (1st string) and E (6th string) of an acoustic guitar?

Do the parents serve a major influence in choosing their children’s career? The statement is a poser which triggers manifold opinion. However, I personally believe that parental opinion do act as an anchorage in children’s decision making. For instance, my father is a sound engineer and spends almost entire working hour in listening and recording tracks and my mother is a professional singer. I was brought up in an environment encircled with musical instrument and other forms of art. Consciously or Unconsciously, I inherited their passion and dedication towards the art, culture and music.

Since childhood, I was mesmerised by the tone of the acoustic guitar which my father used to play at night. String instruments have always influenced my interest in the cultural world. Finally, it was in my third standard when I started acquiring a formal training in playing acoustic guitar. After playing guitar since an age of nine years the only problem while playing guitar is again its most mesmerising factor – the tone. Guitars have to be tuned properly almost every time it requires to be played. Due to lack of formal education on working mechanism of a guitar as well as string theory, till now, I thought the reason of detuning being loosening of keys of guitar. However, currently, after learning oscillation and harmonic motion, I learnt that there exist a relation between frequency and temperature. This theory has given born to a curiosity in my mind.

After contacting with one of our family friend settled in Boston, United States, I came across the fact that guitar strings are tightened with different intensity during the winters than that of summers in Boston. Establishing with the fact I learnt from Boston, I wanted to dive deeper into the effect of atmospheric temperature in the tune of guitar. I studied a few journals and articles on temperature dependency of frequency, and found a few equations where temperature affects the frequency of any oscillating object; but failed to find any direct information about the effect on temperature on specifically guitar strings. To find the answer to my question, I have decided to explore and find the answer to my question through this assessment.

Acoustic guitar is one of the most popular used musical instrument and also one of the most widely used musical instrument available in today’s world. This is because of its mesmerising soothing tone, portability, light weighted structure and also financial affordability.

Structurally, an acoustic guitar is divided into several parts as shown in Figure 1.

In an acoustic guitar six strings are present, which is tied by the keys at one end and by the pins placed at Bridge on the other side. The keys help to loosen or strengthen the intensity of tightness of the individual string. When the intensity of strain increases, every string pitch also increases and vice versa. The keys are there to tune the strings to it’s fundamental tone accurately.

Each string in guitar comprise a fundamental tone. From Figure 2, normal six string guitar corresponds to the fundamental tones of E, B, G, D, A, E from 1^st to 6^th string respectively.

Each string has its fundamental frequency which determines the tone of each string. The fundamental frequency of each string of an acoustic guitar is shown below in a tabular form.

As per Vincenzo Galilei , propagation velocity of a wave produced by a vibrating string is directly proportional to the square root of tension applied on the string. It is also inversely proportional to the square root of linear density of the string.

The mathematical expression of the above mentioned relation is stated below -

\(v ∝ \sqrt{\frac{T}{\mu}}\)

Here, v = the propagation velocity of wave,

T = string tension,

µ = linear density of string.

Experimentally, it has been found that,

\(v \sqrt{\frac{T}{\mu}}\)

Linear density = the mass of any body of unit length.

Hence, the previous equation can be written as follows -

\(v = \sqrt{\frac{T}{\frac{m}L}}\)  .........(1)

Here, m =string mass

L = string length.

Frequency is inversely proportional to the wavelength of the wave, considering that the wave velocity stays constant.

The mathematical expression,

\(f =\frac{v}{λ}\) .........(2)

Where, f = wave frequency,

λ = the wavelength of the wave,

v = wave velocity.

From equation (1) and (2 -

\(f = \frac{1}{\lambda}\sqrt{\frac{T}{\frac{m}L}}\) ........(3)

The law states that the fundamental harmonic nodes produced by a vibration lies at both ends of the string. So, if the length of string be L, then the wavelength of the vibration will be 2L. Thus, from equation (3), we concluded:

\(f = \frac{1}{2L}\sqrt{\frac{T}{\frac{m}L}}\) .........(4)

Length of any solid objects increases with an increase in temperature. This occurs because the kinetic energy of the object molecules increased. The mathematical expression for increase in length -

∆L = L₀ × α × ∆T ... ... ... (5)

Here, ∆L = the change in length of string,
α = co-efficient of thermal expansion,
∆T = change in temperature.

Co-efficient of thermal expansion is defined as the increase in length of a solid object of unit length for an increase in temperature of unity.

Let, the initial length of a string be L₀, the coefficient of thermal expansion of the material of string be α, the fundamental frequency of the string be f₀, mass of the string be m, tension in the string be  T_e, therefore, the expression of the frequency can be expressed as:

\(f_{o}=\frac{1}{2L_{o}}\sqrt{\frac{T _{e}}{\frac{m}{L_o}}}\)

Let, the string is heated and the change in string temperature to be ∆T. Therefore, the final length (L_f) of the string can be expressed as:

L_f  = L₀ + L₀ × α × ∆T

=> L_f  = L₀(1 + α∆T)

Therefore, the expression of final frequency (f_f) of the string can be expressed as -

 \(f_{f}=\frac{1}{2L_{f}}\sqrt{\frac{T _{e}}{\frac{m}{L_f}}}\)

 \(=> f_f  =\frac{1}{2L_{o}(1\,+\, α∆T)}\sqrt{\frac{T _{e}}{\frac{m}{L_o(1\,+\,α∆T)}}}\)           (6)

In the above equation (6), physical parameters such as initial length of the string (L₀), coefficient of linear expansion of string (α), mass of string (m), initial fundamental frequency of string (f₀) are constant parameters and do not depend on temperature or change in temperature.

Thus, the above equation (7) can be expressed as -

\(f_f  ∝ \frac{1}{∆T}\sqrt{T_{e}∆T}\)           (7)

Moreover, tension would be varied with temperature. This is because, as the temperature increases (increase in the magnitude of change of temperature), the string length would increase and hence the tension in the guitar string would decrease.

 \(Te ∝\frac{1}{∆T}\)          (8)

Combining equation (7) and (8):

 \(f_f  ∝\frac{1}{∆T} \sqrt\frac{{1}}{∆T}×∆T\)

\(∴ f_f  ∝ \frac{1}{∆T}\)           (9)

From equation (9), it can be stated that with an increase in the magnitude of change in temperature due to increase in temperature, the final frequency of the string will decrease.

Therefore, the change in frequency (∆f) of string due to increase in temperature can be expressed as -

∆f  =  f_f   −  f₀

As the final frequency (f_f) decrease with an increase in temperature, the difference in frequency would increase with an increase in temperature.

Therefore, mathematically, it could be expressed as -

∆f ∝ − ∆T

There is a negative sign before the notation of change in temperature because with an increase in temperature, the frequency would decrease.

In this exploration, two acoustic guitar strings, i.e., 3^rd string (G string) and 4^th string (D string) determines the temperature effect on change of guitar string frequency. These two strings are considered for the exploration as the E(1^st) and B strings are too thin, so they might melt during the experiment as temperature increased to 60°C. On the other hand, the E(6^th) and A strings are too thick, so often they gather rust layer and other impurities on them that cannot be separated without an experimental process. A different reason of not taking E(6^th) and A string was the resistance of the string. As these strings are thick they have higher cross sections. Thus their resistance decreases. Thus, it will take a lot of current to reach the required heat to make the temperature more than 35°C. 3^rd and 4^th strings, as they are placed at the centre of the guitar, their observed nature in these two strings would have been carried out in all the other strings.

A power supply (DC battery) was used and the principle of Joule’s Law was followed to heat to increase the guitar string temperature because this is the easiest way to increase the guitar string temperature without taking out the strings from the guitar. If the guitar string temperature is increased by any other method that will also increase the temperature of metallic parts and the body of the guitar. This would have created higher errors in experimental observations.

Thereafter, for increase in every 5°C in temperature of each of the two guitar strings, the guitar string was plucked with same intensity using a plectrum and the same position on the string and the frequency was measured using a mobile application.

In the research journal titled as – ‘Effect of temperature changes on the function of the electric guitar’ by Baškarić, Tomislav, et al. in the Proceedings of TEAM 2014 (2014): 472, it was concluded that guitar strings should be manufactured by material having less coefficient of thermal expansion to reduce the extent of effective increase or decrease in length of string controlled by the atmospheric temperature.

It is assumed that with an increase in temperature, the frequency of each string will decrease. This is because due to linear expansion of solid string, the length of string will increase with an increase in its temperature initiated by the increase in atmospheric temperature as shown in equation no. (5). Consequently, it will be loosened without even altering the keys of guitar. In guitar, if the strings are loosened, the pitch of the tone of string decreases. As pitch is directly proportional to frequency, the string frequency will decrease with increase in its temperature. Furthermore, with an increase in length of the string, the fundamental frequency of the string will decrease as discussed in equation no (6).

Temperature of string

The temperature of the string was considered to be the independent variable in this exploration. The temperature was increased from 25°C to 55°C with a regular interval of 5°C using an electric circuit. The temperature was varied over such a range because temperature usually varies over this range in India and other subtropical countries. The temperature of the string was increased by passing electric current through the string. When current passed through any string or wire, according to Joule’s Law of Heating, heat is dissipated by the string. As a result, the temperature of the string was increased.

Frequency of string

The frequency of tone produced by the string at each trial of temperature was the dependent variable in this exploration. Itwas because, frequency is the only driving parameter which was responsible for detuning of any string of guitar. The frequency was measured using a functional microphone in Google Science Journal Application on an android.

• It was made sure that the G string and D string of the guitar by adjusting the turning pegs until the frequency is reached.
• It was made sure that gloves have been worn.
• It was made sure that the aforementioned control variables are being controlled as well as possible and the surroundings are clear of any noise.
• Infrared radiation thermometer was used to measure the initial temperature of the guitar string.

• An acoustic guitar with brass strings connected in the guitar was taken.
• The strings were rubbed using a clean cloth to remove dirt and impurities from it.
• A DC Voltage Supply (battery) of 12 V is obtained and the two ends were connected with two different crocodile to crocodile probes.
• The free end of both the crocodiles probes were connected with two ends of the G string.
• It was verified that the metal parts of the crocodile probe was not in touch with any other string.
• The battery was turned on and the temperature of the string was being constantly observed using the infrared thermometer.
• The temperature was increased from 30°C to 50°C. At every interval of 5°C, the battery has been turned off and the G string was played once using a plectrum.
• The frequency of the string was measured using Google Science Journal Application on a smartphone with a fully functional microphone and noted.
• Once the procedure was completed, the battery is turned off and the guitar was left for some time so that the temperature of the strings fall back to the room temperature.
• After that, the same procedure has been repeated two times.
• Finally, the entire process is repeated for D string.

• Gloves were worn to avoid finger burns if unknowingly it touched the heated strings.
• Wore rubber gloves to stop ourselves from saving us being electrocuted because the experimental procedure involves use of electricity.
• An apron was worn throughout the experiment to prevent ourselves from snapped guitar string.
• For precaution, a fire extinguisher was kept at our side because the guitar body, which was made of woods, was flammable. The fire could have been started from a short circuit because we were using current.

• The experiment was undertaken only with two guitar strings (G string and D string). This was to reduce the cost of experiment.
• As an android application was used to string frequency readings instead of an expensive instrument, it decreased experiment cost.
• Only a 12 V battery was used without using high volt batteries. It has not only decreased the experimental procedure, but also allowed to have a slower temperature increase rate. With batteries with higher voltage, higher amount of heat would have been passed through and temperature would have increased at a much faster rate. This would have made the experiment difficult to process.

• Electricity or Joule’s Law of Heating was used to increase guitar string temperature. During the proceedings of the experiment no fossil fuel has been burnt or emission of greenhouse gas has been observed

Raw Data Table

Processed Data Table

Sample Calculation

Change in Temperature = 30.0°C − 25.0°C = 5.0°C

Change in Frequency = 196.0 − 186.6 = 9.3 Hz

Impact in Uncertainty

Uncertainty in change in temperature

∆T_change = T_initial − T_final

\(\frac{ΔT_{change}}{T_{change}} = \frac{ΔT_{initial}}{T_{initial}}\ +\ \frac{ΔT_{final}}{T_{final}}\)

= 0.1 + 0.1 =  ± 0.2°C

Uncertainty in change in frequency of string

∆f_change = f_initial − f_final

\(\frac{Δf_{change}}{_{f_{change}}} = \frac{Δf_{_{initial}}}{f_{initial}}\ +\ \frac{Δf_{final}}{f_{final}}\)

= 0.1 + 0.1 =  ± 0.2°C

Graphical Analysis

Analysis

In Graph 8, the variation in the fundamental frequency of G string (3^rd string) of an acoustic guitar with respect to change in temperature from room temperature (25.0°C) has been obtained. The change in temperature expressed in °C has been plotted along the X – Axis and the change in frequency of the string measured in Hz has been plotted along the Y – Axis. It has been observed that when the temperature of string was changed by 30.0°C, the change in frequency increased from 9.3Hz to 13.5Hz. An overall increasing trend has been obtained between the change in frequency with respect to the change in temperature. The equation of trend obtained in the graph is shown below -

y = 0.1493x + 8.8757

In the above equation, y indicates the change in frequency measured in Hz and x indicates the change in temperature measured in °C.

Despite having a single outlier in the graph at x = 15, the remaining data points were lying on the trend satisfying the existence of the obtained trend. Moreover, the strength of the trend could be explained by such a high value of regression correlation coefficient (= 0.95).

Real Life Significance and Scientific Justification

With an increase in temperature, the change in frequency increases. This should be because of increase in length of the string caused due to thermal expansion of solid. As the length of the string increased, wavelength of the wave generated by the string during playing the string also increased. As a result, the string frequency decreases. Thus, if the temperature of the string was increased due to change in atmospheric temperature or any other direct or indirect reasons, the frequency of the string decreases and the sound generated by the string becomes dull (low pitched as frequency is directly proportional to pitch). Thus the guitar string got detuned.

Raw Data Table

Processed Data Table

Graphical Analysis

Analysis

In Figure 10, the variation in the fundamental frequency of D string (4^th string) of an acoustic guitar with respect to change in temperature from room temperature (25.0°C) has been obtained. The change in temperature expressed in °C has been plotted along the X – Axis and the change in frequency of the string measured in Hz has been plotted along the Y – Axis. It has been observed that when the temperature of string is changed by 30.0°C, the change in frequency increased from 7.5Hz to 13.9Hz. An overall increasing trend has been obtained between the change in frequency with respect to the change in temperature. The equation of trend obtained in the graph is shown below:

y = 0.248x + 6.66

In the above equation, y indicates the change in frequency measured in Hz and x indicates the change in temperature measured in °C.

Despite having a single outlier in the graph at x = 10, the remaining data points are lying on the trend satisfying the existence of the obtained trend. Moreover, the strength of the trend can be explained by such a high value of regression correlation coefficient (= 0.98).

Real Life Significance and Scientific Justification

With an increase in temperature, the change in frequency increases. This should be because of increase in length of the string caused due to thermal expansion of solid. As the string length increased, wavelength of the wave generated by the string during playing the string also increased. As a result, string frequency decreased. Thus, if the temperature of the string was increased due to change in atmospheric temperature or any other direct or indirect reasons, the frequency of the string decreases and the sound generated by the string becomes dull (low pitched as frequency is directly proportional to pitch). Thus the guitar string got detuned.

How does atmospheric temperature ranging between 25°C to 55°C with a regular interval of 5°C affect the frequency of G (3^rd string) and D (4^th string) of an acoustic guitar?

The frequency of string of an acoustic guitar decreases linearly with an increase in temperature.

• It was observed that when the temperature of G string of an acoustic guitar changed by 25.0°C to 55.0°C, the frequency of the string decreased from 186.6 H_z to 182.4 Hz.
• It is observed that when the temperature of G string of an acoustic guitar was changed by 30.0°C (starting from room temperature 25°C), the change in frequency increased from 9.3Hz to 13.5Hz.
• The equation of trend obtain between the change in frequency (in Hz) (y)with respect to change in temperature (in °C) (x)is shown below: y = 0.1493x + 8.8757.
• The strength of the trend obtained for G string of the acoustic guitar as determined by regression coefficient was given by: r² = 0.95.
• It is observed that when the temperature of D string of an acoustic guitar was changed by 25.0°C to 55.0°C, the frequency of the string decreased from 139.7 Hz to 133.3 Hz.
• It is observed that when the temperature of D string of an acoustic guitar changed by 30.0°C (starting from room temperature 25°C), the change in frequency increased from 7.5Hz to 13.9 Hz.
• The equation of trend obtain between the change in frequency (in Hz) (y) with respect to change in temperature (in °C) (x)was shown below: y = 0.248x + 6.66.
• The strength of the trend obtained for G string of the acoustic guitar as determined by regression coefficient was given by: r² = 0.98.

• The string temperature was measured by IR thermometer. It has eliminated physical contact from the exploration. This in turn gave us minimum errors in the readings and increased exploration reliability.
• The almost zero values of Standard deviation meant that the trial readings were close to each other, which meant a reliable exploration.
• The laws of Physics explained the graphs shown in the experiment.
• The instrument taken, an acoustic guitar was an accurate representation as the dimensions of the strings are universally consented.
• The process of heating was uniform and efficient throughout the strings.
• Three trials of dependent variable are taken for every value of independent variable, and the mean value of the trials were taken. This process is known as Method of Triangulation. It reduced the error or discrepancy present in the observed data which in turn made a coherent exploration.

Methodological Limitation

• The assumption was that, only the length of the string was increasedwith an increase in temperature. However, it might not be the case. The string cross section area can also expand due to the application of the heat.
• Another assumption was that, once the power supply was turned off; there was all the interaction between room temperature and string temperature stopped. However, it is obvious that there would be some heat exchange between them.

Rather than testing the fundamental frequency of the guitar string when it is attached to the guitar, it can be tested in isolation so that the expansion or contraction of wood can be controlled and prevent it from affecting the frequency of the bass guitar string.

The experiment can be observed as follows: The two strings, i.e., 2^nd string and 3^rd string will be used to determine the effect of temperature in change in frequency of strings of bass guitar. A power supply (DC battery) was used and to raise the temperature of guitar strings the principle of Joule’s Law was followed to heat them, as after removing the strings from the guitar, that is the most suitable way to raise the temperature of guitar strings. On the other hand, if any other ways of raising temperature of guitar string are used, it will raise the temperature of the body of the guitar and other metallic parts which will cause many errors in our experimental findings and observations.

Thereafter, for increase in every 5°C in temperature of each of the two guitar strings, the guitar string is plucked with same intensity using a plectrum and the same position on the string and the frequency was measured using a mobile application.

Therefore, the research question of the exploration could be framed as follows: “How does atmospheric temperature ranging between 25°C to 60°C with a regular interval of 5°C affect the frequency of 2^nd and 3^rd string of a bass guitar obtained using the principles of Joule’s Law of Heating?”

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